Hyperbolicity, irrationality exponents and the eta invariant

TL;DR

This paper investigates the remainder term in the semiclassical eta invariant formula on contact manifolds, linking it to Reeb flow recurrence sets and demonstrating improvements for specific flow types.

Contribution

It establishes a general control of the remainder term via recurrence set volumes and provides new bounds for Anosov and elliptic Reeb flows based on flow dynamics.

Findings

Logarithmic improvement for Anosov Reeb flows

Bounds related to irrationality measures for elliptic flows

Control of the remainder term by recurrence set volumes

Abstract

We consider the remainder term in the semiclassical limit formula for the eta invariant on a metric contact manifold, proving in general that it is controlled by volumes of recurrence sets of the Reeb flow. This particularly gives a logarithmic improvement of the remainder for Anosov Reeb flows, while for certain elliptic flows the improvement is in terms of irrationality measures of corresponding Floquet exponents.